Vibrational stability improvement of a mirror system using active mass damping

Active damping with internal absolute velocity feedback was implemented to damp the angular vibration of a mirror holder for free-electron laser beam transportation.


Introduction
Advanced X-ray light sources have become very important tools for research in modern science applications due to their high brightness, short pulse and good coherence.Compared with traditional synchrotron radiation sources, free-electron lasers (FELs) are characterized by higher peak brightness, shorter pulse width and better coherence (Pellegrini & Sto ¨hr, 2003;Zhao & Feng, 2018), and are hence considered to be a new generation of light sources (Georgescu, 2020).As part of the Zhangjiang scientific facility group of advanced light sources, Shanghai High repetitioN rate XFEL and Extreme light facility (SHINE) is expected to achieve ultra-high peak brightness and average brightness, high repetition rate and femtosecond-level ultrafast pulses, as well as nanoscale ultrahigh spatial resolution and femtosecond level ultrafast time resolution.
X-ray beam transportation is up to 1 km long, in order to take full advantage of the surface smoothness of the beamline optical devices (Siewert et al., 2019), as well as the pointing stability -the angular demands of mirrors in the transportation should be less than 50 nrad.For such high requirements, disturbance from the tunnel floor, motion devices and cooling devices (Li et al., 2011) cannot be ignored.Most of the disturbances will be attributed to structural vibrations of mechanical structures which will then affect the beam transportation stability (Houghton et al., 2021).
In order to improve the stability of instruments at advanced light sources, passive vibration isolation (Mangra et al., 1996;Van Vaerenbergh et al., 2008) is generally used in engineering: the influence of vibration on the instrument is improved by optimizing the physical design of the mechanical structure.However, passive vibration isolation is a poor suppression effect for low-frequency vibrations.Furthermore, it is difficult to increase damping in a vacuum, and damping materials age quickly (Liu et al., 2020).
With the development of microelectronics technology, research is increasingly focused on the active vibration reduction method, which can effectively damp structural resonances and low-frequency vibrations (Guoping et al., 2004), with good flexibility.For active vibration control, the actuator arrangement of active vibration isolation is often limited by the actual mechanical structure, while the actuator position of active mass damping (AMD) (Gonzalez Diaz, 2007) is independent.However, AMD could cause serious resonance at its natural frequency (required to be lower than the controlled band), called the 'spillover effect', so the problem caused by the 'spillover effect' is intolerable for allfrequency sensitive applications.
In this work, an internal absolute velocity feedback (IAVF) is proposed for AMD to reduce the influence of the spillover effect, and to improve the modal vibration damping performance of the controlled structure.The angular pitch vibration attenuation performance was demonstrated on a mirroradjusting mechanism for the SHINE project.

Model of active mass damping
To attenuate structural vibrations, AMD is introduced.For simplicity, a two-degrees-of-freedom model is built.The principle of AMD is shown in Fig. 1, where m 1 is the mass of the controlled structure and m 2 is the mass of the active mass damper (AMDer).The dynamic equation of the system is where X 1 (s), c 1 and k 1 are the Laplace transform of the absolute displacement, damping coefficient and stiffness of the controlled structure, respectively; X 2 (s), c 2 and k 2 are the Laplace transform of the displacement, damping coefficient and stiffness of the AMDer, respectively.F is the secondary force generated by a voice coil motor (VCM), expressed as where K f , R and � are the force constant, coil resistance and electrical time constant of the VCM, respectively (the back electromotive force was ignored here).U is the driving voltage of the VCM computed by the feedback controller, which, in the ordinary AMD, is described as and the PID control algorithm G c1 is simply expressed as To reduce the spillover effect and improve the modal vibration attenuation performance, an IAVF is introduced to increase the active damping of the AMDer.A control diagram is illustrated in Fig. 2, according to which the driving voltage of the VCM became where is part of the control algorithm for IAVF.According to all formulas above, with and the vibrational transfer function from ground to controlled object, G p1 , and that from U 0 to controlled object vibration, G s1 , are expressed as Two-degrees-of-freedom model of the active mass damping system.

Figure 2
Control diagram of AMD with IAVF.
and the vibrational transfer function from the ground to the internal mass of the AMDer, G p1 , and that from U 0 to the internal mass of the AMDer, G s1 , are expressed as In particular, the definitions of G p10 , G s10 , G p20 , G s20 in Fig. 2 are, respectively, Considering the closed-loop transfer function by Fig. 2 (ignoring the dynamic characteristic of sensor), the control performance reflects the transmissibility from the ground to the controlled structure with AMD.In this work, velocity feedback was mainly adopted, since it is effective for multi-mode attenuation.
According to formula ( 12), the open-loop frequency response (OLFR) of the unit-gain velocity feedback, with the parameters listed in Table 1 is shown in Fig. 3(a).There are two resonant peaks in the amplitude-frequency response curve: one with lower natural frequency and lower amplitude, which is caused by the AMDer, the other with higher natural frequency and higher amplitude, which is from the stability-performance formula proposed by Gonzalez Diaz ( 2007), where � k0 = jG OLFR ð j! k Þj=jG OLFR ð j! a Þj, the performance of modal vibration attenuation could be improved by IAVF.
According to formula (17), the transmissibility of AMD is shown in Fig. 4. The active damping effect at the natural frequency of the controlled structure is significantly improved as the gain [g max = 1/|G OLFR (j! a )|] is raised; however, the spillover effect at the natural frequency of the AMDer increases.With the IAVF, it is obvious that the increase at AMD resonance is suppressed, while the same vibration attenuation performance is achieved at the resonance of the controlled structure.As a cost, the frequency range of the spillover effect is widened due to the active damping effect of IAVF.

Introduction of a test setup
To test the performance of the AMDer with IAVF, a prototype of the AMDer was built, installed and tested at the mirror regulating mechanism for the SHINE offset mirror.

Introduction of the mirror regulating system
The whole adjusting mechanism was designed as shown in Fig. 5.A detailed description has been given by Liu et al. (2023).A double-layer granite base was used for coarse pitch angle adjustment and horizontal translation, and the mirror holder was supported by three vertical adjustments to meet the vertical, roll and yaw adjustments.Since the pitch angle is

Figure 5
The prototype of the offset mirror adjustment system for the SHINE beamline.
the most critical angle that we need to deflect or focus the beam, a fine pitch adjustment was designed by flexures on the mirror holder driven by a piezo actuator.
The modal of the system was measured with accelerometers (model 393B04, PCB), hammer (type INV9311) and DASP (version 11) software (http://www.coinv.com/product/59).The related pitch angular modes are shown in Fig. 6.The first pitch angular mode at 12 Hz and the third pitch angular mode at 24 Hz are due to stiffness differences on the ends of the mirror holder, the second angular mode at 18 Hz is attributed to the coupling of the translation, while the fourth mode at 160 Hz is from the fine pitch adjustment of the mirror holder.The absolute pitch angle vibration was calculated from the angular velocity by placing two velocity gauges (model 941B, produced by the Institute of Engineering Mechanics of China Earth-quake Bureau) at a distance of 800 mm apart.The pitch angular spectrum is shown in Fig. 7, while the ground vibration in the vertical direction is shown in Fig. 8.The angular vibration below 5 Hz was attributed to ground excitations.The 14 Hz and 17 Hz vibrations were found to be external machine excitations from the ground, as they could be detected across the whole laboratory, and the spectra are rather narrow.The vibrations around 12 Hz, 18 Hz and 25 Hz are attributed to the structural modal vibrations of the adjustments.The fourth mode from the fine pitch adjustment was seen to be of little affect because of its high Eigen frequency.

Introduction of the AMDer
A 3D model of the AMDer is shown in Fig. 9; the actuator is a voice coil from Akribis (type AVM-60.25).The magnetic part is fixed to the mass, while the coil part is fixed to the base.The mass is connected to metal pieces serving as springs, and a velocity gauge (model 941B mentioned above) is also fixed to the mass for the IAVF.The signal acquisition and feedback controller is implemented in the NI cRIO-9042 RT/FPGA system: the PID controller was implemented at a 25.6 kHz sample rate.The key parameters of the AVM-60.25-typeVCM and 941B-type velocity gauge are listed in Table 2. Two velocity gauges (model 941B) are fixed to the mirror holder to measure the angular velocity as Figure 7 Vibration spectra of the offset mirror without the AMDer.

Figure 8
Ground vibration spectra in the vertical direction.

Figure 9
Structure of the AMDer.
where _ X 11 and _ X 12 are the velocities measured by the two gauges, and L = 413 mm is the distance between the two gauges.The whole test setup is shown in Fig. 10.For easy mounting, the velocity gauges are fixed on the mirror holder rather than on the mirror dummy, because the pitch angular vibration on the mirror and the holder are almost the same by our measurement.

Characteristic rest
Before the performance test, the frequency response characteristics G OLFR were obtained by driving the actuator with a sinusoidal frequency scanning voltage and recording the amplitude ratio at each excitation frequency !.As shown in Fig. 11, these resonant peaks agree well with the three modals contributing to the main vibration shown in Fig. 7; the peak at 6 Hz is caused by the AMDer.
In the absence of extra damping, the resonance of the AMD is only damped by the properties of the material; since the springs are made of stainless steel, the damping ratio was very small, so the open-loop response amplitude G s ( j! a ) is too high to implement a high-gain velocity feedback, which will limit the performance of modal vibration attenuation.With IAVF ( p 2 = À 4.58 � 10 À 5 mm s À 1 V À 1 ), G s ( j! a ) decreases significantly while the change of amplitude at the natural frequency of all the controlled structure resonance G s ( j! k ) is negligible, which indicates that the modal vibration attenuation performance is improved.
The damping ratio was also fitted according to the waveform by the impulse response of the AMDer as shown in Fig. 12.By fitting the wave with the following equation, the damping rate, �, of the resonance of the AMDer increases from 0.03 to 0.19 with IAVF.

Performance test and discussions
The pitch angular vibrations of the mirror holder measured by the two velocity gauges for different conditions are shown in Fig. 13.Spectra and the accumulated RMS are shown in the top and bottom panels, respectively, of Fig. 14.When the A comparison of the secondary path frequency response with and without IAVF.

Figure 10
Test setup of the pitch angular vibration damping.

Table 2
Parameters of the actuator, sensor and electronics.AMDer with IAVF is off, there is an additional peak at about 6 Hz compared with the spectrum in Fig. 7 without the AMDer, which is due to the natural frequency of the AMDer.
Owing to the spillover effect, the damping factor with normal AMD is limited due to vibration increase at 6 Hz; the performance is then also limited.However, when the AMDer with IAVF is active, the peaks at around 6 Hz, 12 Hz, 18 Hz and 25 Hz are significantly decreased.Even at 14 Hz, the vibration caused by some external excitation is also attenuated from 26 nrad to 15 nrad.As a result, the RMS above 1 Hz of angular vibration of the mirror holder drops from 47 nrad to 27 nrad.

Conclusion
Active mass damping with IAVF was introduced to attenuate the structural angular vibration of a mirror system for FEL beamlines.Performance tests demonstrated that the IAVF can reduce the spillover effect and further improve the structural vibration attenuation performance.By comparing the angular vibrations with and without the active mass damping, the angular vibration of the mirror system was improved from 47 nrad to 27 nrad above 1 Hz in RMS.

Figure 12
Comparison of the damping ratio with and without IAVF.

Figure 3
Figure 3Open-loop frequency response of unity gain exterior velocity feedback.

Figure 4
Figure 4Transmissibility for AMD without IAVF (a) and with IAVF (b).

Figure 6
Figure 6 Modal shape at (a) 12 Hz, (b)18 Hz and (c) 25 Hz of the mirror regulating system.

Table 1
Parameters of the AMD model.